Remark 1 In [1]}, the extremal inequality was proved in two steps: Firstly, Gaussian random vector \(Z_{1}\)  is transformed into \(\tilde{Z}_{1}\)  via enhancement argument [2]}, which is degraded to \(Z_{2}\) . Secondly, a monotone path centered approach is invoked to show that Gaussian optimality of the auxiliary optimization problem on arbitrary distributed random vectors. Alternatively, Geng and Nair introduced a doubling trick to obtain Liu-Viswanath extremal inequality directly in [3]}. Motivated by the doubling trick, we construct an appropriate monotone path in the tensorized probability space, and show the Gaussian optimality without enhancement argument.

